Abstract
It is well known that the rate of convergence of S-iteration process introduced by Agarwal et al. (pp. 61-79) is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce some Newton-like algorithms to solve the non-linear operator equation in Banach space setting. We study the semi-local as well as local convergence analysis of our algorithms. The rate of convergence of our algorithms are faster than the modified Newton method.Mathematics Subject Classification 2010: 49M15; 65K10; 47H10.
Highlights
Let D be an open convex subset of a Banach space X and F be a Fréchet differentiable operator at each point of D with values in a Banach space Y
Theorem 1.1 Let F be a Fréchet differentiable operator defined on an open convex subset D of a Banach space X with values in a Banach space Y
Theorem 1.2 Let F be a Fréchet differentiable operator defined on an open convex subset D of a Banach space X with values in a Banach space Y
Summary
Let D be an open convex subset of a Banach space X and F be a Fréchet differentiable operator at each point of D with values in a Banach space Y. Lemma 2.4 [9]Let F be a Fréchet differentiable operator defined on an open convex subset D of a Banach space X with values in a Banach space Y. Lemma 2.6 [[17], Theorem 9.4.2] Let F be a Fréchet differentiable operator defined on an open convex subset D of a Banach space X with values in a Banach space Y. Before studying convergence analysis of Algorithm 1.3, we establish the following theorem for existence of a unique solution of operator Equation (1.1). Before presenting local convergence result for Algorithm 1.4, we need the following theorem: Theorem 3.9 Let F be a Fréchet differentiable operator defined on an open convex subset D of a Banach space X with values in a Banach space Y. For x0 Î D and a Î (0, 1), let Fx−∗ 1and F satisfy the
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